Integrality of varifolds in the singular limit of reaction-diffusion equations

By Yoshihiro Tonegawa. 2003

54 citations recorded by the source corpus.

We answer a question posed by Ilmanen on the integrality of varifolds which appear as the singular perturbation limit of the Allen-Cahn equation. We show that the density of the limit measure is integer multiple of the surface constant almost everywhere at almost all time. This shows that limit measures obtained via the Allen- Chan equation and those via Brakke's construction share the same integrality property as well as being weak solutions for the mean curvature flow equation.

Subfields: Applied Mathematics, Computational Theory and Mathematics

View Paper: https://www.semanticscholar.org/paper/dd7c29683e1dbd420472275c357daaf7e52a9cb3

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Applied Mathematics

1. Lami Kim
2. Yoshihiro Tonegawa
3. Kota Kasai
4. Lu Wang
5. J. Bernstein

In Computational Theory and Mathematics

1. Kota Kasai
2. Lami Kim
3. Yoshihiro Tonegawa

Citations in the corpus

• On the mean curvature flow of grain boundaries (2015, methodology): “As a different track, there are other methods such as elliptic regularization [29] and phase field approximation via the Allen–Cahn equation [28, 44] to obtain rigorous global-in-time existence results of Brakke flow.”
• A general regularity theory for weak mean curvature flow (2011, methodology): “In addition the second author [33] proved that the limit varifolds are integral modulo division by a surface energy constant for a.”